Electric Displacement Distributions Research Articles

Analytical solutions to Mode I penny-shaped crack problems in two-dimensional hexagonal quasicrystals with piezoelectric effect

The paper studies a penny-shaped crack in an infinite three-dimensional body of two-dimensional hexagonal quasicrystal media with piezoelectric effect. The crack surfaces are applied combined electric and normal phonon loadings. Such a Model I crack problem is transformed into a mixed boundary value problem in the upper half-space, which is analytically solved using Fabrikant's potential theory method. The boundary integral-differential equations governing Model I crack problems are presented for two-dimensional hexagonal piezoelectric quasicrystals. The normal phonon displacement discontinuity and electric potential discontinuity across crack surfaces are taken as the unknown variables of boundary governing equations. Analytical solutions of all field variables are derived not only for the crack plane but also for the full space. Solutions in integral form are provided for the penny-shaped crack under arbitrarily distributed electric and normal phonon loadings. Closed-form solutions in terms of elementary functions are given for concentrated point loadings and uniformly distributed loadings, respectively. Key fracture mechanics parameters, such as crack surface extended displacements (i.e., normal phonon displacement, electric potential), crack tip extended stresses (i.e., normal phonon stress, electric displacement) distribution, and corresponding extended stress intensity factors, are clearly derived. Numerical results are utilized to verify the present analytical solutions and graphically illustrate the distribution of phonon-phason-electric coupling fields around the crack. The present solution can serve as a benchmark for both experimental and numerical investigations.

Extended Theory and Analysis of Potential Applications of Gauss's Law

This article analyzes and discusses the extended theory of Gauss's law and its potential applications in different fields. In the theoretical foundation section, the article elaborates on the definition, theory, and formulas of the traditional Gauss's law, providing mathematical derivations. Additionally, it reviews the applications of Gauss's law in electrostatics and gravitational fields, including aspects such as electric field flux, electric potential displacement, gravitational field, and mass distribution. In the potential application analysis section, the article explores the potential applications of Gauss's law in fields such as communications, acoustics, and medicine The expanded theories and potential applications of Gauss's theorem provide us with a unified mathematical framework at the macroscopic level for describing and explaining physical phenomena in different fields. By applying Gauss's theorem and its extended forms, we can better understand and explore various phenomena in the natural world, applying them to acoustics, medicine, and scientific research, thus advancing technological development and addressing practical problems.

Performance investigation and parameter identification of inverse variable cross-section energy harvester

An important issue in traditional energy harvesters is that the best performance of the device is limited to the uneven longitudinal strain and electric displacement distributions. One solution to this challenge is to incorporate of variable cross-section design and rotational motion. In this study, we propose a new rotational inverse complex piezoelectric energy harvester featuring two beams for ultra-low-frequency rotation applications. A comprehensive electromechanical model is developed based on the extended Hamilton's principle, and the analytical models are extracted for vibration response and electric performance under ultra-low-frequency excitations. Closed-form analytical expressions of the rotational frequency, electrical load resistance, and centrifugal coefficient are derived to give in-depth insights into the influence mechanism of the centrifugal softening effect for the extreme conditions of RL→0 and RL→∞. Moreover, we propose a parameter identification method to recognize optimal electrical load resistance for ultra-low-frequency rotational scenarios in the absence case of experimental equipment. Based on the validated theoretical model, parametric studies are conducted to investigate the effects of the structural parameters on the dynamic behaviors and output electric performance, it can be concluded that the appropriation selection of the structural parameters can change the dynamic behaviors of the harvester and enhance the energy harvesting performance. Besides, the prototype of the proposed harvester was designed and fabricated and experiments were carried out. Both analytical simulation and experimental results show that the proposed inverse energy harvester has an excellent harvesting performance at the ultra-low-frequency rotational excitations, and the output voltage of the proposed inverse energy harvester is 12% larger than that of a traditional inverse energy harvester.

Effect of Adhesive Layer Thickness and Slant Angle on Piezoelectric Bonded Joints

Piezoelectric materials have found their parts in numerous manufacturing applications such as transducers and sensors. Piezoelectric material shows anisotropy; in addition, its elastic field and electric field are integrated. The analysis of piezoelectric materials has obtained great interest among researchers with the development of smart structures. It is significant to explain the distribution of stress and electric displacement fields along the bonded edge. It has never been obvious how stress singularity and electric displacement fields are distributed at the vertex of piezoelectric dissimilar material joints. Stress and electric displacement distribution near the vertex along the interface of piezoelectric boned joints are investigated in this present research. Numerical analysis of piezoelectric dissimilar material joints is carried out by using Abaqus FEA software. From the numerical analysis, it is observed that stress, displacement, electric potential, electric displacement field evolvement along the interface edge rises with the increment of adhesive layer thickness and slant angle. So, a thin adhesive layer thickness and small slant angle are more reliable for operation.

Reflection and transmission of elastic waves through nonlocal piezoelectric plates sandwiched in two solid half-spaces

A nonlocal stress expansion Legendre orthogonal polynomial method is proposed to study the reflection and transmission of elastic waves through piezoelectric nanoplates sandwiched in two solid half-spaces. The proposed method avoids employing the relationship between local stress and nonlocal stress to construct boundary conditions. The reflection and transmission of plane waves and out-plane waves are studied separately with considering the electrical open circuit boundary. The convergence of presented method is discussed. The correctness of presented method is verified by comparing with the published data of piezoelectric nanoplates immersed in water, which can be considered as a simplified situation sandwiched in two solid half spaces. In addition, the nonlocal effect on the wave mode conversion, stress and electric displacement distribution, and piezoelectricity are discussed. Results show that the nonlocal effect reconstructs the wave mode conversion, and enhances the influence of the piezoelectricity on the critical angle.

Dispersion Waves in a Submerged Thermo-Piezoelectric Membrane

Abstract Wave propagation in a thermo piezoelectric membrane immersed in an infinite fluid medium is discussed using three-dimensional linear theory of elasticity and thermos piezoelectricity. Three displacement potential functions are introduced to uncouple the equations of motion, heat and electric conduction equations. The frequency equations are obtained for longitudinal and flexural modes at the solid fluid interfacial boundary conditions. The numerical results are analyzed for PZT-4 material and the computed stress, strain, electric displacement and temperature distribution are presented in the form of dispersion curves and its characteristics are studied.

Modeling and interpreting particle shape dependence of electric displacement distribution in the piezoelectric polymer composites

A finite element–based code was developed to capture the effect of particle shape on the electromechanical properties of three typical piezoelectric polymer composites. The electrical energy density and electric displacement distribution patterns were modeled by putting particles with different shapes into the model. To evaluate the reliability of investigations, differential equations were solved using Mori–Tanaka and finite element methodologies, where model predictions were appropriately consistent with each other regardless of particle shape and volume fraction of minor phase. Polymer composites filled with continuous fiber revealed similar electromechanical features irrespective of the piezoelectric nature of polymers. On the other hand, strain- and stress-induced electrical displacement distributions were sensitive to particle shape. The results suggest that densities corresponding to relative strain energy and relative electrical energy of composites containing continuous fiber inclusions are both higher than the analogous values obtained for the systems filled with short fiber and spherical particles. Likewise, the polymer composite with continuous fiber revealed improved electric displacement distribution compared to those filled with the other two types of filler. The truthfulness of the developed model was further ensured by placing multifarious particles into the host polymers, which is of vital importance from application point of view.

Frictional contact problem between a functionally graded magnetoelectroelastic layer and a rigid conducting flat punch with frictional heat generation

ABSTRACTIn this study, we consider the frictional sliding contact problem between a functionally graded magnetoelectroelastic (MEE) layer resting on a perfectly insulated rigid half plane and a perfectly conducting rigid flat punch with frictional heat generation. The punch is subjected to magnetoelectromechanical loads. The graded layer is modeled as a nonhomogeneous medium with a transversely isotropic stress–strain law and an exponential variation of the magnetoelectrothermoelastic properties along the thickness direction. Neglecting inertia effects and assuming a constant friction coefficient, the solution is obtained within the framework of steady-state plane magnetoelectrothermoelasticity under plane strain conditions. The heat equation is first solved using Fourier transform to yield the temperature field in the layer which is then substituted in the MEE governing equations. These equations are solved analytically using the same transform leading to three coupled Cauchy-type singular integral equations in which the main unknowns are the normal contact stress, the electric displacement, and the magnetic induction. These equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement, and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the nonhomogeneity parameter; the friction coefficient; and the elastic, electric, and magnetic coefficients on the surface contact pressure, electric displacement, and magnetic induction distributions for the case of flat punch profile.

Transient thermoelectroelastic response of a piezoelectric material strip with two parallel cracks in arbitrary positions

ABSTRACTThis work is concerned with the thermoelectromechanical fracture behavior of two parallel cracks in arbitrary positions of a piezoelectric material strip under thermal shock loading. The crack faces are supposed to be insulated thermally and electrically. By using both the Laplace transform and the Fourier transform, the thermal and electromechanical problems are reduced to two systems of singular integral equations. The singular integral equations are solved numerically, and a numerical method is then employed to obtain the time-dependent solutions by way of a Laplace inversion technique. The intensity factors versus time for various geometric parameters are calculated and presented in graphical forms. The temperature, stress and electric displacement distributions in a transient state are also included.

The contact problem of a rigid stamp with friction on a functionally graded magneto-electro-elastic half-plane

We consider in this study the frictional sliding contact problem between a functionally graded magneto-electro-elastic material and a perfectly conducting rigid punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the unknowns are the normal contact stress, the electric displacement and the magnetic induction. These integral equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the non-homogeneity parameter, the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat and circular punch profiles.

Transient Thermoelectroelastic Response of a Functionally Graded Piezoelectric Material Strip with Two Parallel Cracks in Arbitrary Positions

In this article, the problem of two parallel cracks in arbitrary positions of a functionally graded piezoelectric material (FGPM) strip is analyzed under transient thermal loading conditions. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the crack faces are supposed to be insulated thermally and electrically. By using both the Laplace transform and the Fourier transform, the thermal and electromechanical problems are reduced to two systems of singular integral equations. The singular integral equations are solved numerically, and a numerical method is then employed to obtain the time dependent solutions by way of a Laplace inversion technique. The intensity factors versus time for various geometric and material parameters are calculated and presented in graphical forms. Temperature change, the stress and electric displacement distributions in a transient state are also included.

SBFEM for fracture analysis of piezoelectric composites under thermal load

This paper extends a semi-analytical technique, the so-called scaled boundary finite element method (SBFEM), to analyze fracture behaviors of piezoelectric materials and piezoelectric composites under thermal loading. In this method, only the boundary is discretized leading to a reduction of the spatial dimension by one. The temperature field in the domain is obtained using the SBFEM and expressed as a series of power functions of the radial coordinate. The resulting stress and electric displacement distribution along the radial direction is represented analytically. This permits the generalized stress and electric displacement intensity factors to be directly evaluated from the solution by following standard stress recovery procedures in the finite element method (FEM). Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique.

On the frictional sliding contact problem between a rigid circular conducting punch and a magneto-electro-elastic half-plane

Abstract The frictional sliding contact problem between a homogeneous magneto-electro-elastic material (MEEM) and a perfectly conducting rigid circular punch subjected to magneto-electro-mechanical loads is considered in this paper. The problem is formulated under plane strain conditions and the resulting plane magneto-electro-elasticity equations are converted analytically using Fourier transform into three coupled singular integral equations in which the main unknowns are the normal contact stress, the electric displacement and the magnetic induction. The main contribution of this work is the derivation of an analytical closed-form solution for the normal contact stresses, electric displacement and magnetic induction distributions. The primary purpose of this investigation is to study the effect of the friction coefficient, the punch radius, the applied magneto-electro-mechanical loadings, the material composition on the contact surface and in-plane surface stresses, electric displacement and magnetic induction distributions for the case of a circular stamp profile.

Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch

This paper focuses on the study of a frictional sliding contact problem between a homogeneous magneto-electro-elastic material (MEEM) and a perfectly conducting rigid flat punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the main unknowns are the normal contact stress, the electric displacement and the magnetic induction. An analytical closed-form solution is obtained for the normal contact stress, electric displacement and magnetic induction distributions. The main objective of this paper is to study the effect of the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat stamp profile.

Electro-elastic fields of piezoelectric materials with an elliptic hole under uniform internal shearing forces

The existing investigations on piezoelectric materials containing an elliptic hole mainly focus on remote uniform tensile loads. In order to have a better understanding of the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole in transversely isotropic piezoelectric materials subjected to uniform internal shearing forces based on the complex potential approach. By solving ten variable linear equations, the analytical solutions inside and outside the hole satisfying the permeable electric boundary conditions are obtained. Taking PZT-4 ceramic into consideration, numerical results of electro-elastic fields along the edge of the hole and axes, and the electric displacements in the hole are presented. Comparison with stresses in transverse isotropic elastic materials shows that the hoop stress at the ends of major axis in two kinds of material equals zero for the various ratios of major to minor axis lengths; If the ratio is greater than 1, the hoop stress in piezoelectric materials is smaller than that in elastic materials, and if the ratio is smaller than 1, the hoop stress in piezoelectric materials is greater than that in elastic materials; When it is a circle hole, the shearing stress in two materials along axes is the same. The distribution of electric displacement components shows that the vertical electric displacement in the hole and along axes in the material is always zero though under the permeable electric boundary condition; The horizontal and vertical electric displacement components along the edge of the hole are symmetrical and antisymmetrical about horizontal axis, respectively. The stress and electric displacement distribution tends to zero at distances far from the elliptical hole, which conforms to the conclusion usually made on the basis of Saint-Venant’s principle. Unlike the existing work, uniform shearing forces acting on the edge of the hole, and the distribution of electro-elastic fields inside and outside the elliptic hole are considered.

Wave propagation in multilayered piezoelectric spherical plates

This paper proposes an improvement of the Legendre polynomial series method to solve the harmonic wave propagation in multilayered piezoelectric spherical plates, which are used in point-focusing transducers. The conventional Legendre polynomial method can deal with the multilayered structures only when the material properties of two adjacent layers do not change significantly and cannot obtain correctly normal stress and normal electric displacement shapes unlike the proposed improved orthogonal polynomial approach which overcomes these drawbacks. Detailed formulations are given to highlight its differences from the conventional Legendre polynomial approach. Through the comparisons of numerical results given by an exact solution (obtained from the reverberation-ray matrix method), and by the conventional polynomial approach and the improved polynomial approach, the validity of the proposed approach is illustrated. The influences of the radius-to-thickness ratio on dispersion curves, stress and electric displacement distributions are discussed. It is found that three factors determine the distribution of mechanical energy and electric energy at higher frequencies: radius-to-thickness ratio, wave speed, and position of the component material.

Three-Dimensional Piezothermoelastic Stress of a Finite Functionally Graded Cylindrical Shell with Piezoelectric Layer

Three-dimensional piezothermoelastic solutions for a finite functionally graded cylindrical shell with piezoelectric layer are carried out in this paper. The cylindrical shell is simply supported at four end edges and is subjected to axisymmetric thermomechanical loads. The piezoelectric layers are polarized along radial direction as a sensor. The material properties are assumed to be temperature independent and radially dependent but are assumed to be homogeneous in each layer; the variables are expanded in Fourier series to satisfy the boundary conditions and multilayer approach is used. Numerical results of mullite/molybdenum functionally graded cylindrical shell are presented; the temperature change, stresses, electric potential, and electric displacement distributions are given and briefly discussed.

Three-dimensional piezo-thermo-mechanical analysis for a finite cylindrical composite panel composed of cross-ply and piezoelectric laminate

The exact three-dimensional piezo-thermo-mechanical solution for a finite cylindrical composite panel composed of cross-ply and piezoelectric laminate is presented in this article. The panel is simply supported at four end edges and is subjected to thermo-mechanical loadings on its top and bottom surfaces, respectively. The piezoelectric layers are polarized along radial direction as a sensor. The variables are expanded layerwise in Fourier series to satisfy the boundary conditions at the simply supported ends. As an example, a cross-ply laminated cylindrical panel made of alumina fiber reinforced aluminum composite, associated with a piezoelectric material of crystal class mm2 is calculated. Furthermore, some numerical results for the temperature change, the displacement, stresses, electric potential, and electric displacement distributions are shown in figures and briefly discussed.

Wave propagation in the circumferential direction of general multilayered piezoelectric cylindrical plates

The Legendre polynomial approach has been proposed to solve wave propagation in multilayered flat plates and functionally graded structures for more than ten years, but it can deal with a multilayered plate only when the material properties of two adjacent layers do not change significantly. In this paper, an improvement of the Legendre polynomial approach is proposed to solve wave propagation in what, from now on, we will call general multilayered piezoelectric cylindrical plates, to mean indifferently with or without very dissimilar materials. Detailed formulations are given to highlight the differences from the conventional Legendre polynomial approach. Through numerical comparisons among the exact solution (from the reverberation-ray matrix), the conventional polynomial approach, and the improved polynomial approach, the validity of the proposed approach is illustrated. Then, the influences of the radius-to-thickness ratio on the dispersion curves, the stress, and electric displacement distributions are discussed. It is shown that the conventional orthogonal polynomial approach cannot obtain correct continuous normal stress and normal electric displacement shapes, unlike the improved orthogonal polynomial approach, which overcomes these drawbacks. It is also found that three factors determine the distribution of mechanical energy and electric energy at higher frequencies: the radius-to-thickness ratio, the wave speed of component material, and the position of the component material.

Transient Thermoelectromechanical Response of a Piezoelectric Strip with Two Parallel Cracks of Different Lengths

This work is concerned with the thermoelectromechanical fracture behavior of two parallel cracks of different lengths in a piezoelectric material strip under thermal shock loading. The crack faces are supposed to be insulated thermally and electrically. By using both the Laplace transform and the Fourier transform, the thermal and electromechanical problems are reduced to two systems of singular integral equations, respectively, which are solved numerically. A numerical method is employed to obtain the time dependent solutions by way of a Laplace inversion technique. The intensity factors versus time for various geometric parameters are calculated and presented in graphical forms. Temperature change, the stress and electric displacement distributions in a transient state are also included.